For a finite p-group
G
, the lower exponent-p central series (briefly lower p-central series) of
G
is a descending series
(
P
j
(
G
)
)
j
≥
0
of characteristic subgroups of
G
, defined recursively by
(
1
)
P
0
(
G
)
:=
G
and
P
j
(
G
)
:=
[
P
j
−
1
(
G
)
,
G
]
⋅
P
j
−
1
(
G
)
p
, for
j
≥
1
.
Since any non-trivial finite p-group
G
>
1
is nilpotent, there exists an integer
c
≥
1
such that
P
c
−
1
(
G
)
>
P
c
(
G
)
=
1
and
c
l
p
(
G
)
:=
c
is called the exponent-p class (briefly p-class) of
G
. Only the trivial group
1
has
c
l
p
(
1
)
=
0
. Generally, for any finite p-group
G
, its p-class can be defined as
c
l
p
(
G
)
:=
min
{
c
≥
0
∣
P
c
(
G
)
=
1
}
.
The complete lower p-central series of
G
is therefore given by
(
2
)
G
=
P
0
(
G
)
>
Φ
(
G
)
=
P
1
(
G
)
>
P
2
(
G
)
>
⋯
>
P
c
−
1
(
G
)
>
P
c
(
G
)
=
1
,
since
P
1
(
G
)
=
[
P
0
(
G
)
,
G
]
⋅
P
0
(
G
)
p
=
[
G
,
G
]
⋅
G
p
=
Φ
(
G
)
is the Frattini subgroup of
G
.
For the convenience of the reader and for pointing out the shifted numeration, we recall that the (usual) lower central series of
G
is also a descending series
(
γ
j
(
G
)
)
j
≥
1
of characteristic subgroups of
G
, defined recursively by
(
3
)
γ
1
(
G
)
:=
G
and
γ
j
(
G
)
:=
[
γ
j
−
1
(
G
)
,
G
]
, for
j
≥
2
.
As above, for any non-trivial finite p-group
G
>
1
, there exists an integer
c
≥
1
such that
γ
c
(
G
)
>
γ
c
+
1
(
G
)
=
1
and
c
l
(
G
)
:=
c
is called the nilpotency class of
G
, whereas
c
+
1
is called the index of nilpotency of
G
. Only the trivial group
1
has
c
l
(
1
)
=
0
.
The complete lower central series of
G
is given by
(
4
)
G
=
γ
1
(
G
)
>
G
′
=
γ
2
(
G
)
>
γ
3
(
G
)
>
⋯
>
γ
c
(
G
)
>
γ
c
+
1
(
G
)
=
1
,
since
γ
2
(
G
)
=
[
γ
1
(
G
)
,
G
]
=
[
G
,
G
]
=
G
′
is the commutator subgroup or derived subgroup of
G
.
The following Rules should be remembered for the exponent-p class:
Let
G
be a finite p-group.
R
- Rule:
c
l
(
G
)
≤
c
l
p
(
G
)
, since the
γ
j
(
G
)
descend more quickly than the
P
j
(
G
)
.
- Rule: If
ϑ
∈
H
o
m
(
G
,
G
~
)
, for some group
G
~
, then
ϑ
(
P
j
(
G
)
)
=
P
j
(
ϑ
(
G
)
)
, for any
j
≥
0
.
- Rule: For any
c
≥
0
, the conditions
N
◃
G
and
c
l
p
(
G
/
N
)
=
c
imply
P
c
(
G
)
≤
N
.
- Rule: Let
c
≥
0
. If
c
l
p
(
G
)
=
c
, then
c
l
p
(
G
/
P
k
(
G
)
)
=
min
(
k
,
c
)
, for all
k
≥
0
, in particular,
c
l
p
(
G
/
P
k
(
G
)
)
=
k
, for all
0
≤
k
≤
c
.
Parents and descendant trees
The parent
π
(
G
)
of a finite non-trivial p-group
G
>
1
with exponent-p class
c
l
p
(
G
)
=
c
≥
1
is defined as the quotient
π
(
G
)
:=
G
/
P
c
−
1
(
G
)
of
G
by the last non-trivial term
P
c
−
1
(
G
)
>
1
of the lower exponent-p central series of
G
. Conversely, in this case,
G
is called an immediate descendant of
π
(
G
)
. The p-classes of parent and immediate descendant are connected by
c
l
p
(
G
)
=
c
l
p
(
π
(
G
)
)
+
1
.
A descendant tree is a hierarchical structure for visualizing parent-descendant relations between isomorphism classes of finite p-groups. The vertices of a descendant tree are isomorphism classes of finite p-groups. However, a vertex will always be labelled by selecting a representative of the corresponding isomorphism class. Whenever a vertex
π
(
G
)
is the parent of a vertex
G
a directed edge of the descendant tree is defined by
G
→
π
(
G
)
in the direction of the canonical projection
π
:
G
→
π
(
G
)
onto the quotient
π
(
G
)
=
G
/
P
c
−
1
(
G
)
.
In a descendant tree, the concepts of parents and immediate descendants can be generalized. A vertex
R
is a descendant of a vertex
P
, and
P
is an ancestor of
R
, if either
R
is equal to
P
or there is a path
(
5
)
R
=
Q
0
→
Q
1
→
⋯
→
Q
m
−
1
→
Q
m
=
P
, where
m
≥
1
,
of directed edges from
R
to
P
. The vertices forming the path necessarily coincide with the iterated parents
Q
j
=
π
j
(
R
)
of
R
, with
0
≤
j
≤
m
:
(
6
)
R
=
π
0
(
R
)
→
π
1
(
R
)
→
⋯
→
π
m
−
1
(
R
)
→
π
m
(
R
)
=
P
, where
m
≥
1
.
They can also be viewed as the successive quotients
Q
j
=
R
/
P
c
−
j
(
R
)
of p-class
c
−
j
of
R
when the p-class of
R
is given by
c
l
p
(
R
)
=
c
≥
m
:
(
7
)
R
≃
R
/
P
c
(
R
)
→
R
/
P
c
−
1
(
R
)
→
⋯
→
R
/
P
c
+
1
−
m
(
R
)
→
R
/
P
c
−
m
(
R
)
≃
P
, where
c
≥
m
≥
1
.
In particular, every non-trivial finite p-group
G
>
1
defines a maximal path (consisting of
c
=
c
l
p
(
G
)
edges)
(
8
)
G
≃
G
/
1
=
G
/
P
c
(
G
)
→
π
(
G
)
=
G
/
P
c
−
1
(
G
)
→
π
2
(
G
)
=
G
/
P
c
−
2
(
G
)
→
⋯
ending in the trivial group
π
c
(
G
)
=
1
. The last but one quotient of the maximal path of
G
is the elementary abelian p-group
π
c
−
1
(
G
)
=
G
/
P
1
(
G
)
≃
C
p
d
of rank
d
=
d
(
G
)
, where
d
(
G
)
=
dim
F
p
(
H
1
(
G
,
F
p
)
)
denotes the generator rank of
G
.
Generally, the descendant tree
T
(
G
)
of a vertex
G
is the subtree of all descendants of
G
, starting at the root
G
. The maximal possible descendant tree
T
(
1
)
of the trivial group
1
contains all finite p-groups and is exceptional, since the trivial group
1
has all the infinitely many elementary abelian p-groups with varying generator rank
d
≥
1
as its immediate descendants. However, any non-trivial finite p-group (of order divisible by
p
) possesses only finitely many immediate descendants.
p-covering group, p-multiplicator and nucleus
Let
G
be a finite p-group with
d
generators. Our goal is to compile a complete list of pairwise non-isomorphic immediate descendants of
G
. It turns out that all immediate descendants can be obtained as quotients of a certain extension
G
∗
of
G
which is called the p-covering group of
G
and can be constructed in the following manner.
We can certainly find a presentation of
G
in the form of an exact sequence
(
9
)
1
⟶
R
⟶
F
⟶
G
⟶
1
,
where
F
denotes the free group with
d
generators and
ϑ
:
F
⟶
G
is an epimorphism with kernel
R
:=
ker
(
ϑ
)
. Then
R
◃
F
is a normal subgroup of
F
consisting of the defining relations for
G
≃
F
/
R
. For elements
r
∈
R
and
f
∈
F
, the conjugate
f
−
1
r
f
∈
R
and thus also the commutator
[
r
,
f
]
=
r
−
1
f
−
1
r
f
∈
R
are contained in
R
. Consequently,
R
∗
:=
[
R
,
F
]
⋅
R
p
is a characteristic subgroup of
R
, and the p-multiplicator
R
/
R
∗
of
G
is an elementary abelian p-group, since
(
10
)
[
R
,
R
]
⋅
R
p
≤
[
R
,
F
]
⋅
R
p
=
R
∗
.
Now we can define the p-covering group of
G
by
(
11
)
G
∗
:=
F
/
R
∗
,
and the exact sequence
(
12
)
1
⟶
R
/
R
∗
⟶
F
/
R
∗
⟶
F
/
R
⟶
1
shows that
G
∗
is an extension of
G
by the elementary abelian p-multiplicator. We call
(
13
)
μ
(
G
)
:=
dim
F
p
(
R
/
R
∗
)
the p-multiplicator rank of
G
.
Let us assume now that the assigned finite p-group
G
≃
F
/
R
is of p-class
c
l
p
(
G
)
=
c
. Then the conditions
R
◃
F
and
c
l
p
(
F
/
R
)
=
c
imply
P
c
(
F
)
≤
R
, according to the rule (R3), and we can define the nucleus of
G
by
(
14
)
P
c
(
G
∗
)
=
P
c
(
F
)
⋅
R
∗
/
R
∗
≤
R
/
R
∗
as a subgroup of the p-multiplicator. Consequently, the nuclear rank
(
15
)
ν
(
G
)
:=
dim
F
p
(
P
c
(
G
∗
)
)
≤
μ
(
G
)
of
G
is bounded from above by the p-multiplicator rank.
As before, let
G
be a finite p-group with
d
generators.
Proposition. Any p-elementary abelian central extension
(
16
)
1
→
Z
→
H
→
G
→
1
of
G
by a p-elementary abelian subgroup
Z
≤
ζ
1
(
H
)
such that
d
(
H
)
=
d
(
G
)
=
d
is a quotient of the p-covering group
G
∗
of
G
.
For the proof click show on the right hand side.
In particular, an immediate descendant
H
of
G
is a p-elementary abelian central extension
(
17
)
1
→
P
c
−
1
(
H
)
→
H
→
G
→
1
of
G
, since
1
=
P
c
(
H
)
=
[
P
c
−
1
(
H
)
,
H
]
⋅
P
c
−
1
(
H
)
p
implies
P
c
−
1
(
H
)
p
=
1
and
P
c
−
1
(
H
)
≤
ζ
1
(
H
)
,
where
c
=
c
l
p
(
H
)
.
Definition. A subgroup
M
/
R
∗
≤
R
/
R
∗
of the p-multiplicator of
G
is called allowable if it is given by the kernel
M
/
R
∗
=
ker
(
ψ
∗
)
of an epimorphism
ψ
∗
:
G
∗
→
H
onto an immediate descendant
H
of
G
.
An equivalent characterization is that
1
<
M
/
R
∗
<
R
/
R
∗
is a proper subgroup which supplements the nucleus
(
18
)
(
M
/
R
∗
)
⋅
(
P
c
(
F
)
⋅
R
∗
/
R
∗
)
=
R
/
R
∗
.
Therefore, the first part of our goal to compile a list of all immediate descendants of
G
is done, when we have constructed all allowable subgroups of
R
/
R
∗
which supplement the nucleus
P
c
(
G
∗
)
=
P
c
(
F
)
⋅
R
∗
/
R
∗
, where
c
=
c
l
p
(
G
)
. However, in general the list
(
19
)
{
F
/
M
∣
M
/
R
∗
≤
R
/
R
∗
is allowable
}
,
where
G
∗
/
(
M
/
R
∗
)
=
(
F
/
R
∗
)
/
(
M
/
R
∗
)
≃
F
/
M
, will be redundant, due to isomorphisms
F
/
M
1
≃
F
/
M
2
among the immediate descendants.
Two allowable subgroups
M
1
/
R
∗
and
M
2
/
R
∗
are called equivalent if the quotients
F
/
M
1
≃
F
/
M
2
, that are the corresponding immediate descendants of
G
, are isomorphic.
Such an isomorphism
φ
:
F
/
M
1
→
F
/
M
2
between immediate descendants of
G
=
F
/
R
with
c
=
c
l
p
(
G
)
has the property that
φ
(
R
/
M
1
)
=
φ
(
P
c
(
F
/
M
1
)
)
=
P
c
(
φ
(
F
/
M
1
)
)
=
P
c
(
F
/
M
2
)
=
R
/
M
2
and thus induces an automorphism
α
∈
A
u
t
(
G
)
of
G
which can be extended to an automorphism
α
∗
∈
A
u
t
(
G
∗
)
of the p-covering group
G
∗
=
F
/
R
∗
of
G
. The restriction of this extended automorphism
α
∗
to the p-multiplicator
R
/
R
∗
of
G
is determined uniquely by
α
.
Since
α
∗
(
M
/
R
∗
)
⋅
P
c
(
F
/
R
∗
)
=
α
∗
[
M
/
R
∗
⋅
P
c
(
F
/
R
∗
)
]
=
α
∗
(
R
/
R
∗
)
=
R
/
R
∗
, each extended automorphism
α
∗
∈
A
u
t
(
G
∗
)
induces a permutation
α
′
of the allowable subgroups
M
/
R
∗
≤
R
/
R
∗
. We define
P
:=
⟨
α
′
∣
α
∈
A
u
t
(
G
)
⟩
to be the permutation group generated by all permutations induced by automorphisms of
G
. Then the map
A
u
t
(
G
)
→
P
,
α
↦
α
′
is an epimorphism and the equivalence classes of allowable subgroups
M
/
R
∗
≤
R
/
R
∗
are precisely the orbits of allowable subgroups under the action of the permutation group
P
.
Eventually, our goal to compile a list
{
F
/
M
i
∣
1
≤
i
≤
N
}
of all immediate descendants of
G
will be done, when we select a representative
M
i
/
R
∗
for each of the
N
orbits of allowable subgroups of
R
/
R
∗
under the action of
P
. This is precisely what the p-group generation algorithm does in a single step of the recursive procedure for constructing the descendant tree of an assigned root.
Capable p-groups and step sizes
A finite p-group
G
is called capable (or extendable) if it possesses at least one immediate descendant, otherwise it is terminal (or a leaf). The nuclear rank
ν
(
G
)
of
G
admits a decision about the capability of
G
:
G
is terminal if and only if
ν
(
G
)
=
0
.
G
is capable if and only if
ν
(
G
)
≥
1
.
In the case of capability,
G
=
F
/
R
has immediate descendants of
ν
=
ν
(
G
)
different step sizes
1
≤
s
≤
ν
, in dependence on the index
(
R
/
R
∗
:
M
/
R
∗
)
=
p
s
of the corresponding allowable subgroup
M
/
R
∗
in the p-multiplicator
R
/
R
∗
. When
G
is of order
|
G
|
=
p
n
, then an immediate descendant of step size
s
is of order
#
(
F
/
M
)
=
(
F
/
R
∗
:
M
/
R
∗
)
=
(
F
/
R
∗
:
R
/
R
∗
)
⋅
(
R
/
R
∗
:
M
/
R
∗
)
=
#
(
F
/
R
)
⋅
p
s
=
|
G
|
⋅
p
s
=
p
n
⋅
p
s
=
p
n
+
s
.
For the related phenomenon of multifurcation of a descendant tree at a vertex
G
with nuclear rank
ν
(
G
)
≥
2
see the article on descendant trees.
The p-group generation algorithm provides the flexibility to restrict the construction of immediate descendants to those of a single fixed step size
1
≤
s
≤
ν
, which is very convenient in the case of huge descendant numbers (see the next section).
We denote the number of all immediate descendants, resp. immediate descendants of step size
s
, of
G
by
N
, resp.
N
s
. Then we have
N
=
∑
s
=
1
ν
N
s
. As concrete examples, we present some interesting finite metabelian p-groups with extensive sets of immediate descendants, using the SmallGroups identifiers and additionally pointing out the numbers
0
≤
C
s
≤
N
s
of capable immediate descendants in the usual format
(
N
1
/
C
1
;
…
;
N
ν
/
C
ν
)
as given by actual implementations of the p-group generation algorithm in the computer algebra systems GAP and MAGMA.
First, let
p
=
3
.
We begin with groups having abelianization of type
(
3
,
3
)
. See Figure 4 in the article on descendant trees.
The group
⟨
27
,
3
⟩
of coclass
1
has ranks
ν
=
2
,
μ
=
4
and descendant numbers
(
4
/
1
;
7
/
5
)
,
N
=
11
.
The group
⟨
243
,
3
⟩
=
⟨
27
,
3
⟩
−
#
2
;
1
of coclass
2
has ranks
ν
=
2
,
μ
=
4
and descendant numbers
(
10
/
6
;
15
/
15
)
,
N
=
25
.
One of its immediate descendants, the group
⟨
729
,
40
⟩
=
⟨
243
,
3
⟩
−
#
1
;
7
, has ranks
ν
=
2
,
μ
=
5
and descendant numbers
(
16
/
2
;
27
/
4
)
,
N
=
43
.
In contrast, groups with abelianization of type
(
3
,
3
,
3
)
are partially located beyond the limit of computability.
The group
⟨
81
,
12
⟩
of coclass
2
has ranks
ν
=
2
,
μ
=
7
and descendant numbers
(
10
/
2
;
100
/
50
)
,
N
=
110
.
The group
⟨
243
,
37
⟩
of coclass
3
has ranks
ν
=
5
,
μ
=
9
and descendant numbers
(
35
/
3
;
2783
/
186
;
81711
/
10202
;
350652
/
202266
;
…
)
,
N
>
4
⋅
10
5
unknown.
The group
⟨
729
,
122
⟩
of coclass
4
has ranks
ν
=
8
,
μ
=
11
and descendant numbers
(
45
/
3
;
117919
/
1377
;
…
)
,
N
>
10
5
unknown.
Next, let
p
=
5
.
Corresponding groups with abelianization of type
(
5
,
5
)
have bigger descendant numbers than for
p
=
3
.
The group
⟨
125
,
3
⟩
of coclass
1
has ranks
ν
=
2
,
μ
=
4
and descendant numbers
(
4
/
1
;
12
/
6
)
,
N
=
16
.
The group
⟨
3125
,
3
⟩
=
⟨
125
,
3
⟩
−
#
2
;
1
of coclass
2
has ranks
ν
=
3
,
μ
=
5
and descendant numbers
(
8
/
3
;
61
/
61
;
47
/
47
)
,
N
=
116
.
Via the isomorphism
Q
/
Z
→
μ
∞
,
n
d
↦
exp
(
n
d
⋅
2
π
i
)
the quotient group
Q
/
Z
=
{
n
d
⋅
Z
∣
d
≥
1
,
0
≤
n
≤
d
−
1
}
can be viewed as the additive analogue of the multiplicative group
μ
∞
=
{
z
∈
C
∣
z
d
=
1
for some integer
d
≥
1
}
of all roots of unity.
Let
p
be a prime number and
G
be a finite p-group with presentation
G
=
F
/
R
as in the previous section. Then the second cohomology group
M
(
G
)
:=
H
2
(
G
,
Q
/
Z
)
of the
G
-module
Q
/
Z
is called the Schur multiplier of
G
. It can also be interpreted as the quotient group
M
(
G
)
=
(
R
∩
[
F
,
F
]
)
/
[
F
,
R
]
.
I. R. Shafarevich has proved that the difference between the relation rank
r
(
G
)
=
dim
F
p
(
H
2
(
G
,
F
p
)
)
of
G
and the generator rank
d
(
G
)
=
dim
F
p
(
H
1
(
G
,
F
p
)
)
of
G
is given by the minimal number of generators of the Schur multiplier of
G
, that is
r
(
G
)
−
d
(
G
)
=
d
(
M
(
G
)
)
.
N. Boston and H. Nover have shown that
μ
(
G
j
)
−
ν
(
G
j
)
≤
r
(
G
)
, for all quotients
G
j
:=
G
/
P
j
(
G
)
of p-class
c
l
p
(
G
j
)
=
j
,
j
≥
0
, of a pro-p group
G
with finite abelianization
G
/
G
′
.
Furthermore, J. Blackhurst (in the appendix On the nucleus of certain p-groups of a paper by N. Boston, M. R. Bush and F. Hajir ) has proved that a non-cyclic finite p-group
G
with trivial Schur multiplier
M
(
G
)
is a terminal vertex in the descendant tree
T
(
1
)
of the trivial group
1
, that is,
M
(
G
)
=
1
⇒
ν
(
G
)
=
0
.
A finite p-group
G
has a balanced presentation
r
(
G
)
=
d
(
G
)
if and only if
r
(
G
)
−
d
(
G
)
=
0
=
d
(
M
(
G
)
)
, that is, if and only if its Schur multiplier
M
(
G
)
=
1
is trivial. Such a group is called a Schur group and it must be a leaf in the descendant tree
T
(
1
)
.
A finite p-group
G
satisfies
r
(
G
)
=
d
(
G
)
+
1
if and only if
r
(
G
)
−
d
(
G
)
=
1
=
d
(
M
(
G
)
)
, that is, if and only if it has a non-trivial cyclic Schur multiplier
M
(
G
)
. Such a group is called a Schur+1 group.